Is it just that the idea of picking randomly from an infinite set doesn't make any sense? Or is this a problem that one also runs in to when trying to pick a random element of a finite set?
Yes, although what you mean by infinite is important, and there's an issue with your premise - it's (almost certainly) impossible.*
Yes, this is a problem when trying to pick a random element of a (large) finite set.**
Here's a link for more information: https://en.wikipedia.org/wiki/Probability_density_function.
*However, your premise is missing something - you should expect to die before you hear the number. That is to say, if we could create a device (or oracle) such as you describe, a 'stop' button would be essential, else it would be useless after we asked it what number it picked. This also holds true for any finite but sufficiently large set. For similar reasons, there are algorithms which are which work better than any other for 'sufficiently large inputs' whose critics claim would require a computer the size of a galaxy to be useful. They might be exaggerating, but they might not be.
**Let there be a set of integers from 1 to N. If we pick a number at random (each choice as likely as any other) the probability of picking a given number is 1/100 or 1%. The probability of not picking it is 99%, but you have to pick one of the numbers, so it happens. 100*1%=100%
Intuitively, you might expect to get a number close to 50. The expected value is 50.5 after all, as (1+100)/2=50.5.
However, an infinite set (such as you described) has no (finite) average, because it has no greatest element, nor is there a number greater than all elements in it. But, for any sufficiently large set and thus, for any such infinitely large set, the average is a number so long it cannot be heard. So you should be very surprised indeed, if you hear an end to the number the machine says. (Consider returning it for a refund, because it does not perform as advertised, making sure it doesn't rely on a 'perpetual motion machine' to function, or leaving an appropriate review on amazon, so no one else gets swindled.)
In order to construct such a set, merely give a formula for a number N so large you cannot hear it, then specify that the max element of the set is at least twice that (Max>=2N). (This might, unfortunately, depend on how the machine says numbers. If it reads everything like phone numbers, this is kind of easy. 10^10^10^10 or such might do it. If you live for a 100 years, and a number is read aloud at a rate of 1 digit per second then you do not live long enough to hear a 3.2 billion digit number.)
TL:DR; above in bold.